Explicit Analytic Solution of Vibration Equation for large domain by mean of the Elzaki projected Differential Transform Method

5244 | P a g e O c t o b e r 1 3 , 2 0 1 5 Explicit Analytic Solution of Vibration Equation for large domain by mean of the Elzaki projected Differential Transform Method Muhammad Suleman, Qingbiao Wu, T. M. Elzaki,d, a Department of Mathematics Zhejiang University, Hangzhou, Zhejiang, China b Department of Mathematics, Comsat institute of Information Technology, Islamabad, Pakistan c Mathematics Department, Faculty of Sciences and Arts, University of Jeddah, Jeddah-Saudi Arabia d Mathematics Department, Sudan University of Science and Technology Abstract


Introduction
In many problems arises in area of science and engineering for large membranes, vibration analysis plays an important role to determine the properties and behavior of vibrations. Vibration arises in music, acoustics membranes, microphones, speaker and numerous other devices. Human tissues and eardrum also shows vibrational characteristics and hearing aid devices are designed after understanding the vibrational behavior of membranes. Linear combination of the modes of structure can be used to explained vibrations. Alternatively propagation of wave travelling in a membrane structure, vibration can also cause the destruction of membrane structure in engineering, so characteristics of vibration of membrane and its dynamic response under the effect of external force become a great important scientific issue and number of researchers studied propagation, transmission and reflection of vibrations, like Tapaswini and Chakraverty studied non probabilistic solution of vibration equation using ADM [10], Yidrim studied the solution of vibration equation of a large membrane using HPM [7], Mohyud-din and Yildrim studied and analyzed the fractional vibrational equation for large membrane [9], further can studied in literature. In this paper we apply Elzaki projected differential transform method (EPDTM) [1,2,3] to solve the vibration equation and different cases has been discussed, numerical and graphical results are found with the help of Maple. In section 2, basic idea of Elzaki transform and projected differential transform method is explained. Solution of the problem can be studied in section 3 and some results and conclusion are discussed in section 4.

Elzaki Transformation
Elzaki transform was introduced by Tarig. M. Elzaki in [3]. From the classical Fourier integral, Like Sumudu transform, Laplace transform and Fourier transform, Elzaki transform is used to simplify the process of solving ordinary and partial differential equations in the time domain. Mathematical formulation of Elzaki transformation is as follows: For a given function in set A, the constant M must be finite number k1, k2 maybe finite or infinite. Elzaki transform is denoted by E (.) and defined by the integral equation, Where variable "v" is used in the transformation to the factor the variable"t" in the argument of the function.

Solution of Vibration equation using EPDTM
Consider an open disk of radius "x" centered at origin representing a shape of "still" drum head. Due to circular geometry of disk we use cylindrical co-ordinates so the mode of vibration of radially symmetric circular drum having radius "x", then the function ϕ does not depend on angular displacement ""θ"" , so the vibration equation simplifies to the equation Where ϕ(x, t) represents the displacement of finding a particle at the point "x" in the instant t, c is the wave velocity of free vibration. To solve Eq. (5) by Elzaki projected differential transform method, first we apply the Elzaki transform on both [ E now by applying projected differential transform method we have the following equation.
x ( The above series will be convergent for The above series will be convergent for As Case I and III the above series is also convergent for Case V:

  
Similarly, with the help of A3 and B3 we find O c t o b e r 1 3 , 2 0 1 5 The Elzaki projected differential transform method (EPDTM) is very powerful tool in order to find the solution of various linear and nonlinear problems, showing its application for vibration of very large membrane. Elzaki Projected differential transform method can be used to solve the physical and engineering problem both analytically and numerically. EPDTM also gives rapidly converging solutions. Numerical results also show the higher degree of accuracy of method.